A297324 -id:A297324 - cp's OEIS frontend

A297323 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 - j*x^j)^k.

1, 1, 0, 1, -1, 0, 1, -2, -2, 0, 1, -3, -3, -1, 0, 1, -4, -3, 2, -1, 0, 1, -5, -2, 8, 4, 5, 0, 1, -6, 0, 16, 9, 16, 1, 0, 1, -7, 3, 25, 9, 18, -3, 13, 0, 1, -8, 7, 34, 0, 4, -35, 6, 4, 0, 1, -9, 12, 42, -21, -26, -90, -33, -31, 0, 0, 1, -10, 18, 48, -56, -66, -145, -56, -66, -72, 2, 0

Offset: 0

Ilya Gutkovskiy, Dec 28 2017

			G.f. of column k: A_k(x) = 1 - k*x + (1/2)*k*(k - 5)*x^2 - (1/6)*k*(k^2 - 15*k + 20)*x^3 + (1/24)*k*(k^3 - 30*k^2 + 155*k - 150)*x^4 - (1/120)*k*(k^4 - 50*k^3 + 575*k^2 - 1750*k + 624)*x^5 + ...
Square array begins:
  1,  1,   1,   1,   1,   1,  ...
  0, -1,  -2,  -3,  -4,  -5,  ...
  0, -2,  -3,  -3,  -2,   0,  ...
  0, -1,   2,   8,  16,  25,  ...
  0, -1,   4,   9,   9,   0,  ...
  0,  5,  16,  18,   4, -26,  ...

Columns k=0..32 give A000007, A022661, A022662, A022663, A022664, A022665, A022666, A022667, A022668, A022669, A022670, A022671, A022672, A022673, A022674, A022675, A022676, A022677, A022678, A022679, A022680, A022681, A022682, A022683, A022684, A022685, A022686, A022687, A022688, A022689, A022690, A022691, A022692.
Main diagonal gives A297324.
Antidiagonal sums give A299209.
Cf. A266964, A297321, A297325, A297328.

Table[Function[k, SeriesCoefficient[Product[(1 - i x^i)^k, {i, 1, n}], {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten

first(n, k) = my(res = matrix(n, k)); for(u=1, k, my(col = Vec(prod(j=1, n, (1 - j*x^j)^(u-1)) + O(x^n))); for(v=1, n, res[v, u] = col[v])); res \\ Iain Fox, Dec 28 2017

G.f. of column k: Product_{j>=1} (1 - j*x^j)^k.

A297322 a(n) = [x^n] Product_{k>=1} (1 + k*x^k)^n.

Original entry on oeis.org

1, 1, 5, 28, 137, 726, 3896, 21071, 115089, 633007, 3500740, 19448573, 108458924, 606787572, 3404112479, 19142919543, 107874784017, 609021410570, 3443952349385, 19503777943838, 110599636109572, 627924447630011, 3568885868192419, 20304321490356084

Offset: 0

Ilya Gutkovskiy, Dec 28 2017

nonn

Robert Israel, Table of n, a(n) for n = 0..1000

Main diagonal of A297321.

Cf. A297324, A297326, A297329.

f:= proc(n) local k;
coeff(series(mul((1+k*x^k)^n,k=1..n),x,n+1),x,n);
end proc:
map(f, [$0..30]); # Robert Israel, Dec 28 2017

Table[SeriesCoefficient[Product[(1 + k x^k)^n, {k, 1, n}], {x, 0, n}], {n, 0, 24}]

a(n) = A297321(n,n).

a(n) ~ c * d^n / sqrt(n), where d = 5.814548482877687529318372516965305077397562... and c = 0.2563102401728134539247148322678842806264... - Vaclav Kotesovec, Aug 01 2019

A297329 a(n) = [x^n] Product_{k>=1} 1/(1 - k*x^k)^n.

Original entry on oeis.org

1, 1, 7, 37, 219, 1276, 7687, 46551, 285043, 1756243, 10883842, 67751289, 423366831, 2654041235, 16683909711, 105129718102, 663837626163, 4199521413019, 26610335585263, 168864540960165, 1073001606214814, 6826237566223329, 43474472693256491, 277152041235941803

Offset: 0

Ilya Gutkovskiy, Dec 28 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Main diagonal of A297328.

Cf. A297322, A297324, A297326.

Table[SeriesCoefficient[Product[1/(1 - k x^k)^n, {k, 1, n}], {x, 0, n}], {n, 0, 23}]

a(n) = A297328(n,n).

a(n) ~ c * d^n / sqrt(n), where d = 6.51548390811914394587815688142024783108478... and c = 0.2552310487728179222346375591994440863074... - Vaclav Kotesovec, Aug 01 2019

A297326 a(n) = [x^n] Product_{k>=1} 1/(1 + k*x^k)^n.

Original entry on oeis.org

1, -1, -1, -1, 27, -76, 95, -295, 2035, -8119, 22714, -66793, 254223, -988651, 3444055, -11402626, 39248691, -141740051, 511583207, -1798826901, 6256648862, -22054706773, 78889160635, -281698897727, 996551999479, -3520566280801, 12522382445455, -44731559517301

Offset: 0

Ilya Gutkovskiy, Dec 28 2017

sign

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Main diagonal of A297325.

Cf. A297322, A297324, A297329.

f:= proc(n) local k;
coeff(series(mul(1/(1+k*x^k)^n,k=1..n),x,n+1),x,n);
end proc:
map(f, [$0..30]); # Robert Israel, Dec 28 2017

Table[SeriesCoefficient[Product[1/(1 + k x^k)^n, {k, 1, n}], {x, 0, n}], {n, 0, 27}]

a(n) = A297325(n,n).

A298986 a(n) = [x^n] Product_{k>=1} (1 - n*x^k)^k.

Original entry on oeis.org

1, -1, -4, 9, 48, 100, -756, -3479, -1600, 24462, 225900, 364573, -643536, -9251736, -36989316, -32397975, 165039872, 1725828525, 5338814652, 8082713829, -26321848400, -233434232766, -811526778964, -1731126953532, 1151302859712, 23632432765000, 113461901639788, 287935019845749

Offset: 0

Ilya Gutkovskiy, Jan 31 2018

sign

Cf. A073592, A266964, A292132, A297324, A298985, A298987, A298988.

Table[SeriesCoefficient[Product[(1 - n x^k)^k, {k, 1, n}], {x, 0, n}], {n, 0, 27}]

cp's OEIS Frontend

A297323 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 - j*x^j)^k.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

A297322 a(n) = [x^n] Product_{k>=1} (1 + k*x^k)^n.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A297329 a(n) = [x^n] Product_{k>=1} 1/(1 - k*x^k)^n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A297326 a(n) = [x^n] Product_{k>=1} 1/(1 + k*x^k)^n.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A298986 a(n) = [x^n] Product_{k>=1} (1 - n*x^k)^k.

Views

Author

Keywords

Crossrefs

Programs

Mathematica